Question

class work: monday 9/8/25

1. a c b (line segment) with ac=6x+2, cb=x+1, ab=52. x=__, ac=__
2. if e lies between b and c then ____
3. if d is in the interior of ∠abe then ____

diagram: v, a, b, c if m∠avc=72°, m∠aub=5x+5, m∠bvc=8x+2. find x=__, m∠aub=, m∠bvc=__

Explanation:

Problem 1:

Step1: Use segment addition postulate

We know that \( AC + CB = AB \). Given \( AC = 6x + 2 \), \( CB = x + 1 \), and \( AB = 52 \). So, \( (6x + 2)+(x + 1)=52 \).

Step2: Simplify and solve for \( x \)

Combine like terms: \( 7x + 3 = 52 \). Subtract 3 from both sides: \( 7x = 52 - 3 = 49 \). Divide by 7: \( x=\frac{49}{7}=7 \).

Step3: Find \( AC \)

Substitute \( x = 7 \) into \( AC = 6x + 2 \): \( AC = 6(7)+2 = 42 + 2 = 44 \).

If \( E \) lies between \( B \) and \( C \), by the segment addition postulate, \( BE + EC = BC \) (or \( B - E - C \) implies \( BE + EC = BC \)).

Step1: Use angle addition postulate

We know that \( m\angle AUB + m\angle BVC = m\angle AVC \). Given \( m\angle AVC = 72^\circ \), \( m\angle AUB = 5x + 5 \), and \( m\angle BVC = 8x + 2 \). So, \( (5x + 5)+(8x + 2)=72 \).

Step2: Simplify and solve for \( x \)

Combine like terms: \( 13x + 7 = 72 \). Subtract 7 from both sides: \( 13x = 72 - 7 = 65 \). Divide by 13: \( x=\frac{65}{13}=5 \).

Step3: Find \( m\angle AUB \)

Substitute \( x = 5 \) into \( m\angle AUB = 5x + 5 \): \( m\angle AUB = 5(5)+5 = 25 + 5 = 30^\circ \).

Step4: Find \( m\angle BVC \)

Substitute \( x = 5 \) into \( m\angle BVC = 8x + 2 \): \( m\angle BVC = 8(5)+2 = 40 + 2 = 42^\circ \).

Answer:

\( x = 7 \), \( AC = 44 \)

Problem 2: